Categoricity and multidimensional diagrams (Q6566412)

This paper develops the tool of multidimensional diagrams to give a partial answer to \textit{S. Shelah}'s categoricity conjecture [Isr. J. Math. 46, 212--273 (1983; Zbl 0552.03019)]:\N\NConjecture: Let \(\mathbf{K}\) be an abstract elementary class with Löwenheim-Skolem number \(\mathrm{LS}(\mathbf{K})\). If \(\mathbf{K}\) is categorical in some \(\lambda\geq\beth_{(2^{\mathrm{LS}(\mathbf{K})})^+}\), then it is categorical in all \(\lambda'\geq\lambda\).\N\NEarlier results have given a positive answer, assuming a mix of\N\begin{itemize}\N\item[--] non-ZFC axioms (weak general continuum hypothesis, large cardinals);\N\item[--] nice structure of \(\mathbf{K}\) (amalgamation, primes, tameness);\N\item[--] specific properties of \(\lambda\) (successor cardinal, of certain cofinality).\N\end{itemize}\N\NTheorem 14.5 provides a positive answer assuming a proper class of strongly compact cardinals. This improves on \textit{S. Shelah} and \textit{M. Makkai}'s result [Ann. Pure Appl. Logic 47, No. 1, 41--97 (1990; Zbl 0704.03015)] which focuses on the special case that \(\lambda\) is a successor and that \(\mathbf{K}=\mathrm{Mod}(\psi)\) where \(\psi\) is a \(L_{\kappa,\omega}\) sentence for some strongly compact \(\kappa\).\N\NCorollary 14.10 (together with a later improvement by \textit{S. Vasey} [Sel. Math., New Ser. 25, No. 5, Paper No. 65, 51 p. (2019; Zbl 1468.03042)]) also provides a positive answer but assumes the weak general continuum hypothesis and that \(\mathbf{K}\) has amalgamation.\N\NWhile other structural properties of \(\mathbf{K}\) such as primes and tameness are not explicitly assumed, they are derived from the original assumptions using multidimensional diagrams -- the \(n\)-dimensional analog to the 2-dimensional independence notion called \textit{nonforking}. Multidimensional diagrams allow a better transfer of \textit{extension} and \textit{uniqueness} properties across different cardinals. Ultimately, a nice enough independence relation (called \textit{excellent}) on multidimensional diagrams leads to the proof of the categoricity conjecture.\N\NSections 1--7 introduce the basic notions and background results, including nonforking. Sections 8--12 build an independence relation on multidimensional diagrams from nonforking and study its properties. Section 13 proves the existence of primes so as to invoke the categoricity transfer by \textit{S. Vasey} [Math. Log. Q. 64, No. 1--2, 25--36 (2018; Zbl 1521.03076)]. Section 14 contains the main theorems and their variants.\N\NSome minor typos in the paper: On page 2314 Remark 3.14(4), ``reasonable'' should have been ``resolvable''. On page 2365 Definition 13.8(1), the embedding should send \(f(a)=b\).